Tokyo Journal of Mathematics

On the $p$-class Tower of a $\textbf{Z}_{\textrm{p}}$-extension

Ali MOUHIB and Abbas MOVAHHEDI

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Abstract

For a number field $k$ and a prime number $p$, let $k_{\infty}$ be a $\textbf{Z}_{\textrm{p}}$-extension of $k$ and $X_{\infty}(k)$ the Galois group over $k_{\infty}$ of the maximal abelian unramified $p$-extension of $k_{\infty}$. We first give a sufficient condition, bearing on the norm index of units in the layers of $k_{\infty}$, for $X_{\infty}(k)$ to be finite. When the prime $p$ is 2 and $X_{\infty}(k)\simeq \textbf{Z}/2\textbf{Z}\times \textbf{Z}/2\textbf{Z}$, we study the structure of the Galois group of the maximal unramified $p$-extension of $k_{\infty}$, improving on some previous results in the case of quadratic fields.

Article information

Source
Tokyo J. Math., Volume 31, Number 2 (2008), 321-332.

Dates
First available in Project Euclid: 5 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1233844054

Digital Object Identifier
doi:10.3836/tjm/1233844054

Mathematical Reviews number (MathSciNet)
MR2477874

Zentralblatt MATH identifier
1209.11095

Subjects
Primary: 11R23: Iwasawa theory
Secondary: 11R11: Quadratic extensions

Citation

MOUHIB, Ali; MOVAHHEDI, Abbas. On the $p$-class Tower of a $\textbf{Z}_{\textrm{p}}$-extension. Tokyo J. Math. 31 (2008), no. 2, 321--332. doi:10.3836/tjm/1233844054. https://projecteuclid.org/euclid.tjm/1233844054


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References

  • A. Azizi and A. Mouhib, Capitulation des $2$-classes d'idéaux de ${\bf Q}(\sqrt{2},\sqrt{d})$ où $d$ est un entier naturel sans facteurs carrés., Acta Arith., 109 (2003), no. 1, 27–63.
  • E. Benjamin and C. Snyder, Real quadratic number fields with $2$-class group of type $(2,2)$ Math. Scand., 76 (1995), no. 2, 161–178.
  • Conner, P. E. and Hurrelbrink, J., Class number parity, Series in Pure Mathematics, 8. World Scientific Publishing Co., Singapore, 1988.
  • B. Ferrero and L. C. Washington, The Iwasawa invariant $\mu_p$ vanishes for abelian number fields, Ann. of Math. (2), 109 (1979), no. 2, 377–395.
  • T. Fukuda, Remarks on $\Z_p$-extensions of number fields, Proc. Japan Acad. Ser. A, 70 (1994), 264–266.
  • D. Gorenstein, Finite Groups, Second edition. Chelsea Publishing Co., New York, 1980.
  • R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math., 98 (1976), no. 1, 263–284.
  • R. Greenberg, Iwasawa theory–-past and present. Class field theory–-its centenary and prospect (Tokyo, 1998), 335–385, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001.
  • H. Ichimura, Note on the class numbers of certain real quadratic fields, Abh. Math. Sem. Univ. Hamburg, 73 (2003), 281–288.
  • K. Iwasawa, A note on the group of units of an algebraic number field, J. Math. Pures Appl. (9), 35 (1956), 189–192.
  • H. Kisilevsky, Number fields with class number congruent to $4$ mod $8$ and Hilbert's theorem $94$, J. Number Theory, 8 (1976), no. 3, 271–279.
  • S. Kuroda, Über den Dirichletschen Körper, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I., 4 (1943). 383–406.
  • Y. Mizusawa, On Greenberg's conjecture on a certain real quadratic field, Proc. Japan Acad. Ser. A Math. Sci., 76 (2000), no. 10, 163–164.
  • Y. Mizusawa, On the maximal unramified pro-2-extension of $\Z_2$-extension of certain real quadratic fields, J. Number Theory, 105 (2004), no. 2, 203–211.
  • Y. Mizusawa, On the maximal unramified pro-2-extension of $\Z_2$-extension of certain real quadratic fields II, Acta Arith., 119 (2005), no. 1, 93–107.
  • T. Nguyen, Quang Do and M. Lescop, Iwasawa descent and co-descent for units modulo circular units, Pure Appl. Math. Q., 2 (2006), no. 2, 465–496.
  • M. Ozaki, Iwasawa invariants of $p$-extensions of totally real number fields, preprint.
  • M. Ozaki and H. Taya, On the Iwasawa $\lambda\sb 2$-invariants of certain families of real quadratic fields, Manuscripta Math., 94 (1997), no. 4, 437–444.
  • L. Rédei and H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, J. Reine Angew. Math., 170 (1933), 69–74.
  • H. P. F. Swinnerton-Dyer, A brief guide to algebraic number theory, London Mathematical Society Student Texts 50, Cambridge University Press, Cambridge, 2001.